Long time asymptotics for the focusing nonlinear Schrödinger equation in the solitonic region with the presence of high-order discrete spectrum

In this paper, we study the initial value problem for focusing nonlinear Schrödinger (fNLS) equation with non-generic weighted Sobolev data that allows presence of high-order discrete spectrum. More precisely, show how to characterize properties eigenfunctions and scattering coefficients in poles; Further is formulated into an appropriate enlarged RH problem, which transformed a solvable model after series deformations. Finally, obtain asymptotic expansion solution fNLS any fixed space-time cone: S ( x 1 , 2 v ) : = { t ∈ R 0 + [ ] } . Our result verification soliton resolution conjecture solitonic region The leading order term includes pole-soliton whose parameters are affected by soliton-soliton interactions through cone soliton-radiation on continuous error up O − 3 / 4 comes from corresponding ∂ ¯ equation.